About: Classifying space for U(n)

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In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. This space with its universal fibration may be constructed as either 1. * the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or, 2. * the direct limit, with the induced topology, of Grassmannians of n planes. Both constructions are detailed here.

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  • In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. This space with its universal fibration may be constructed as either 1. * the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or, 2. * the direct limit, with the induced topology, of Grassmannians of n planes. Both constructions are detailed here. (en)
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  • In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. This space with its universal fibration may be constructed as either 1. * the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or, 2. * the direct limit, with the induced topology, of Grassmannians of n planes. Both constructions are detailed here. (en)
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  • Classifying space for U(n) (en)
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